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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle a\frac{\partial w}{\partial t}+x\frac{\partial^2w}{\partial x^2}=0$.
1. Particular solutions:\hfill\break
$\displaystyle w_1(x,t)=t-ax(\ln x-1)$,\\
$\displaystyle w_2(x,t)=t^2-2axt(\ln x-1)+a^2x^2(\ln x-\frac 52)$,\\
$\displaystyle w_3(x,t)=t^3-3axt^2(\ln x-1)+3a^2x^2t(\ln x-\frac 52)-\frac {a^3x^3}2(\ln x-\frac {10}3)$,\\
$\displaystyle w_n(x,t)=t^n+\sum_{k=1}^{n}\frac {(-a)^kn!x^kt^{n-k}}{(n-k)!k!(k-1)!}(\ln x-c_k)$,\\
where $ c_1=1$, $\displaystyle c_k=c_{k-1}+\frac 1{k-1}+\frac 1k$, and $k=2,\,3,\,\dots$

2. Particular solutions:\\
$\displaystyle w_1(x,t)=tx-\frac {ax^2}2$,\\
$\displaystyle w_2(x,t)=t^2x-ax^2t+\frac {a^2x^3}6$,\\
$\displaystyle w_3(x,t)=t^3x-\frac {3ax^2t^2}{2}+\frac {a^2x^3t}{2}-\frac {a^3x^4}{4!}$,\\
$\displaystyle w_n(x,t)=t^nx+\sum_{k=1}^{n}\frac {(-a)^kn!x^{k+1}t^{n-k}}{k!(k+1)!(n-k)!}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Statistic information
Submission date:Sun 18 May 2008 18:29
Edits by author:1
Last edit by author:Mon 26 Sep 2011 19:49

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